Lepton Mixing Predictions from $S_4$ in the Tri-Direct CP approach to Two Right-handed Neutrino Models

We perform an exhaustive analysis of all possible breaking patterns arising from $S_4\rtimes H_{CP}$ in a new {\it tri-direct CP approach} to the minimal seesaw model with two right-handed neutrinos, and construct a realistic flavour model along these lines. According to this approach, separate residual flavour and CP symmetries persist in the charged lepton, `atmospheric' and `solar' right-handed neutrino sectors, i.e. we have {\it three} symmetry sectors rather than the usual two of the {\it semi-direct CP approach} (charged leptons and neutrinos). Following the {\it tri-direct CP approach}, we find twenty-six kinds of independent phenomenologically interesting mixing patterns. Eight of them predict a normal ordering (NO) neutrino mass spectrum and the other eighteen predict an inverted ordering (IO) neutrino mass spectrum. For each phenomenologically interesting mixing pattern, the corresponding predictions for the PMNS matrix, the lepton mixing parameters, the neutrino masses and the effective mass in neutrinoless double beta decay are given in a model independent way. One breaking pattern with NO spectrum and two breaking patterns with IO spectrum corresponds to form dominance. We find that the lepton mixing matrices of three kinds of breaking patterns with NO spectrum and one form dominance breaking pattern with IO spectrum preserve the first column of the tri-bimaximal (TB) mixing matrix, i.e. yield a TM1 mixing matrix.

The most minimal version of the seesaw mechanism involves two additional right-handed neutrinos [6,7]. In order to increase predictive power of the two right-handed neutrino seesaw model, various schemes to reduce the number of free parameters have been suggested, such as postulating one [8] or two [7] texture zeros, however the latter models with two texture zero are now phenomenologically excluded for NO [9][10][11]. In the charged lepton diagonal basis, together with a diagonal right-handed neutrino mass matrix, the idea of constrained sequential dominance (CSD) has been proposed, involving a Dirac mass matrix with one texture zero and restricted form of the Yukawa couplings [12]. The CSD(n) scheme [12][13][14][15][16][17][18][19] assumes that the coupling of one right-handed neutrino (called "atmospheric") with ν L is proportional to (0, 1, 1), while the second right-handed neutrino (called "solar") has couplings to ν L proportional to (1, n, n − 2) with positive integer n, where ν L ≡ (ν e , ν µ , ν τ ) T L denote the left-handed neutrino fields. The CSD(n) models generally [12][13][14][15][16][17][18][19] predict a TM1 mixing matrix and normal mass hierarchy with a massless neutrino m 1 = 0 [20]. The predictions for lepton mixing parameters and neutrino mass have been studied for the cases of n = 1 [12], n = 2 [13], n = 3 [14][15][16], n = 4 [17,18] and n ≥ 5 [19]. It turns out that the CSD(3) also called Littlest Seesaw (LS) model can successfully accommodate the experimental data on neutrino masses and mixing angles [14][15][16]. The LS model can yield the baryon asymmetry of the Universe via leptogenesis [21][22][23]. The LS structure can also be incorporated into grand unified models [21,24,25]. In practice the LS model can be achieved by introducing S 4 family symmetry, which is spontaneously broken by flavon fields with particular vacuum alignments governed by remnant subgroups of S 4 [15,16]. Furthermore, from the breaking of A 5 flavor symmetry to different residual subgroups in the charged lepton, atmospheric neutrino and solar neutrino sectors, we can obtain the viable golden LS model which predicts the GR1 lepton mixing pattern [26]. Here GR1 mixing matrix preserves the first column of the golden ratio mixing matrix.
The leptonic CP violation is one of the most urgent questions in neutrino oscillation physics. The indication of CP violation in neutrino sector has been reported by T2K [27] and NOνA [28], and the Dirac CP phase δ CP will be intensively probed experimentally in the forthcoming years. In order to address this question theoretically, non-Abelian discrete flavor symmetry combined with generalized CP symmetry have been widely exploited to explain the lepton mixing angles and to predict CP violating phases . Both flavor symmetry G f and CP symmetry H CP are imposed at the high energy scale, and the full symmetry is G f H CP . In the successful semi-direct CP approach, the original symmetry G f H CP is spontaneously broken down to G l and G ν H ν CP in the charged lepton sector and the neutrino sector at lower energies, respectively.
Recently we extended the above semi-direct CP approach to propose a so-called tri-direct CP approach [58] based on the two right-handed neutrino seesaw mechanism, and a new variant of the LS model is found. In the tri-direct CP approach, the common residual symmetry of the neutrino sector is split into two branches: the residual symmetries G atm H atm CP and G sol H sol CP associated with the "atmospheric" and "solar" right-handed neutrino sectors respectively. An abelian subgroup G l is assumed to preserved by the charged lepton mass matrix and it allows the distinction of three generations. It is the combination of these three residual symmetries that provides a new way of fixing the lepton mixing parameters and neutrino masses in tri-direct CP approach.
In the present work, we shall extend the analysis of the tri-direct CP approach for two righthanded neutrino models considerably, beyond the few examples studied in [58], to an exhaustive model independent analysis of all possible phenomenologically viable lepton flavor mixing patterns which arise from the breaking of the parent symmetry S 4 H CP . The lepton mixing matrix is not restricted to TM1 mixing anymore and the mass ordering of the neutrino masses can be either NO or IO. We shall find eight independent phenomenologically interesting mixing patterns for the case of NO neutrino masses and eighteen independent phenomenologically interesting mixing patterns for the case of IO. The eight breaking patterns for NO are labeled as N 1 ∼ N 8 and the other eighteen for IO are labeled as I 1 ∼ I 18 . For each possible breaking pattern, we numerically analyze the predictions of the mixing parameters, the three neutrino masses and the effective mass in neutrinoless double beta decay. We find that all the four breaking patterns N 1 , N 2 , N 3 and I 5 give rise to TM1 mixing. For the cases of N 5 , I 4 and I 5 , the two columns of the Dirac neutrino mass matrix are orthogonal to each other and consequently the texture of form dominance [59][60][61] is reproduced. Furthermore, we implement the case of N 4 with x = −4 and η = ±3π/4 in an explicit model based on S 4 H CP , the required vacuum alignment needed to achieve the remnant symmetries is dynamically realized. In this model, the absolute value of the first column of PMNS matrix is fixed to be 2 6 37 , 13 74 , 13 74 T . The paper is organized as follows: in section 2, we recall the framework of the tri-direct CP approach to two right-handed neutrino models, and we present the generic procedures of how to derive the lepton flavor mixing and neutrino masses from remnant symmetries in the tri-direct CP approach in a model independent way. In section 3, we perform a model independent analysis of five kinds of phenomenologically viable breaking patterns achievable from the underlying symmetry S 4 H CP in the tri-direct CP approach with NO neutrino masses. In section 4, a general analysis of five kinds of breaking patterns with IO neutrino masses are presented. In section 5, we present a new version of the LS model based on S 4 H CP from the tri-direct CP approach. The vacuum alignment, the LO structure and the NLO corrections of the model are discussed. Section 6 is devoted to our conclusion. The group theory of S 4 and its all abelian subgroups are presented in appendix A. In appendix B, we study the breaking patterns N 6 ∼ N 8 in a model independent way. The analysis of the remaining thirteen kinds of breaking patterns with IO are given in appendix C.

The tri-direct CP approach
In the scenario with a discrete flavor group G f and generalized CP symmetry H CP , G f and H CP should be compatible with each other, and they fulfill the following consistency condition [29,30,62,63] X r ρ * r (g)X † r = ρ r (g ), g, g ∈ G f , X r ∈ H CP , (2.1) where ρ r (g) is the representation matrix of the element g in the irreducible representation r of G f , and X r is the generalized CP transformation matrix of H CP . Moreover, the physically well-defined Figure 1: A sketch of the tri-direct CP approach for two right-handed neutrino models, where the high energy family and CP symmetry G f HCP is spontaneously broken down to Gatm H atm CP in the sector of one of the right-handed neutrinos, and G sol H sol CP in the sector of the other right-handed neutrino, with the charged lepton sector having a different residual flavour symmetry G l .
generalized CP transformations should be class-inverting automorphisms of G f [62]. It requires that the elements g −1 and g in Eq. (2.1) belong to the same conjugacy class of G f . The automorphism in Eq. (2.1) thus implies that the mathematical structure of the group comprising G f and CP is in general a semi-direct product G f H CP [29].
In the present work, we shall perform a comprehensive study of lepton mixing patterns which can be obtained from the flavor group S 4 and CP symmetry in the tri-direct CP approach [58]. In the following, we shall firstly review how the tri-direct CP approach allows us to predict the lepton mixing and neutrino masses are predicted in terms of few parameters. In the tri-direct CP approach, the assumed family and CP symmetry G f × H CP at high energy scale is spontaneously broken down to an abelian subgroup G l which is capable of distinguishing the three generations in the charged lepton sector, and it is broken to G atm H atm CP and G sol H sol CP in the atmospheric and solar neutrino sectors respectively. A sketch of the tri-direct CP approach for two right-handed neutrino models is illustrated in figure 1. In the right-handed neutrino diagonal basis, the effective Lagrangian is given by L = −y l Lφ l E c − y atm Lφ atm N c atm − y sol Lφ sol N c sol − 1 2 x atm ξ atm N c atm N c atm − 1 2 x sol ξ sol N c sol N c sol + h.c. , (2.2) where L stands for the left-handed lepton doublets and E c ≡ (e c , µ c , τ c ) T are the right-handed charged leptons, the flavons ξ atm and ξ sol are standard model singlets, the flavons φ l , φ sol and φ atm can be either Higgs fields or combinations of the electroweak Higgs doublet together with flavons. All the four coupling constants y atm , y sol , x atm and x sol would be constrained to be real if we impose CP as symmetry on the theory. Without loss of generality, we assume that the three generations of left-handed leptons doublets transform as a faithful three-dimensional representation 3 under G f . The residual symmetry G l in the charged lepton sector requires that the hermitian combination m † l m l must be invariant under the action of G l , i.e. ρ † 3 (g l )m † l m l ρ 3 (g l ) = m † l m l , g l ∈ G l , (2.3) where the charged lepton mass matrix m l is defined in the convention l c m l l. The diagonalization matrix of the hermitian combination m † l m l is defined as U l with U † l m † l m l U l = diag(m 2 e , m 2 µ , m 2 τ ). From Eq. (2.3), we find that the unitary matrix U l can be derived from U † l ρ 3 (g l )U l = ρ diag 3 (g l ) , (2.4) where ρ diag 3 (g l ) is a diagonal matrix with entries being three eigenvalues of ρ 3 (g l ). In the atmospheric neutrino sector and the solar neutrino sector, as the residual symmetries contain both flavor symmetry and CP symmetry, then the following restricted consistency conditions should be satisfied X atm r ρ * r (g atm i )(X atm r ) −1 = ρ r (g atm j ), g atm i , g atm j ∈ G atm , X atm r ∈ H atm CP , (2.5a) X sol r ρ * r (g sol i )(X sol r ) −1 = ρ r (g sol j ), g sol i , g sol j ∈ G sol , X sol r ∈ H sol CP . (2.5b) The consistency conditions indicate that the mathematical structure of the residual flavor and CP symmetries is a semi-direct product for i = j and it reduces to a direct product for the case of i = j. The consistency equations in Eqs. (2.5a) and (2.5b) can be used to find the residual CP consistent with the residual flavor symmetries of the atmospheric neutrino and the solar neutrino sectors, respectively. In the atmospheric and solar neutrino sectors, the residual symmetries imply that the vacuum alignments of flavons φ atm and φ sol should be invariant under the symmetries G atm H atm CP and G sol H sol CP respectively, i.e.
ρ r (g atm ) φ atm = φ atm , X atm r φ atm * = φ atm , (2.6a) where φ atm and φ sol denote the vacuum alignments of flavons φ atm and φ sol , respectively. After electroweak and flavor symmetry breaking, the flavons φ l , φ atm , φ sol , ξ atm and ξ sol acquire nonvanishing vacuum expectation values (VEVs). From the Lagrangian in Eq. (2.2), one can read out the neutrino Dirac mass matrix and the heavy Majorana mass matrix, where U a and U s are two constants matrices and they are constituted by the Clebsch-Gordan (CG) coefficients which appear in the contractions y atm Lφ atm N c atm and y sol Lφ sol N c sol , respectively. For the sake of convenience in the following, we shall parameterize the combinations U a φ atm ≡ v atm v φa and U s φ sol ≡ v sol v φs , where v atm and v sol are three dimensional column vectors and they denote the directions of the vacuum alignment, v φa and v φs are the overall scale of corresponding flavons. The light effective Majorana neutrino mass matrix is given by the seesaw formula m ν = −m D m N m T D , then we find m ν takes the from where the overall phase ϕ a is given by ϕ a = arg −y 2 atm v 2 φa /(x atm ξ atm ) , m a = |y 2 atm v 2 φa /(x atm ξ atm )|, m s = |y 2 sol v 2 φs /(x sol ξ sol )| and η = arg −y 2 sol v 2 φs /(x sol ξ sol ) − ϕ a . The overall phase ϕ a can be absorbed into the lepton field and it will always be omitted in the following. For convenience the notation r ≡ m s /m a would be used throughout this paper. If the roles of G atm H atm CP and G sol H sol CP are switched, the two columns of the Dirac mass matrix m D would be exchanged. Thus the same neutrino mass matrix would be obtained if one interchanges y atm with y sol and x atm with x sol .
In the following we shall give the detailed procedures for analyzing the phenomenological predictions of the tri-direct CP approach in a model independent way, and we shall present the generic expressions of lepton mixing matrix and neutrino masses. One can easily check that neutrino mass matrix m ν of Eq. (2.8) satisfies m ν v fix = (0, 0, 0) T , (2.9) with v fix ≡ v atm × v sol , (2.10) where v atm ×v sol denotes the cross product of v atm and v sol . The normalized vector of v fix is defined Eq. (2.9) implies thatv fix is an eigenvector of m ν with zero eigenvalue. As a result, the first (third) column of U ν is determined to bev fix for NO (IO) mass spectrum, where U ν is the diagonalization matrix of m ν with U T ν m ν U ν = diag(0, m 2 , m 3 ) for NO case and U T ν m ν U ν = diag(m 1 , m 2 , 0) for IO case. In order to diagonalize the above neutrino mass matrix, we firstly perform a unitary transformation U ν1 , where unitary matrix U ν1 can take the following form 1 ,v sol =v * fix ×v atm . (2.12) Then the neutrino mass matrix becomes where the expressions of the parameters y, z and w are (2.14) The neutrino mass matrix m ν in Eq. (2.13) can be diagonalized through the standard procedure, as shown in Ref. [32,58], where the unitary matrix U ν2 can be written as We find the light neutrino masses are with m 1 = 0, m 2 = m l , m 3 = m h for NO case and m 1 = m l , m 2 = m h , m 3 = 0 for IO case. The rotation angle θ is determined by . (2.18) It is obvious that sin 2θ is always non-negative. The expressions of the phases ψ, ρ and σ are given by . (2.19) Thus the lepton mixing matrix is determined to be where P l is a generic permutation matrix since the charged lepton masses are not constrained in this approach, and it can take the following six possible forms (2.21) If two mixing matrices are related by the exchange of the second and third rows, we shall only consider one of them. The reason is that the atmospheric mixing angle θ 23 becomes π/2 − θ 23 , the Dirac CP phases δ CP becomes π + δ CP and the other mixing parameters are unchanged after the second and third rows of a PMNS matrix are permuted. We notice that if both NO and IO neutrino mass spectrums can be achieved for a residual symmetry, the lepton mixing matrix of IO can be obtained from the corresponding one of NO by multiplying P 312 from the right side, and the expressions of the parameters y, z and w in m ν are identical in NO and IO cases.
In the present work, we will adopt the standard parametrization of the lepton mixing matrix [64], where c ij ≡ cos θ ij , s ij ≡ sin θ ij , δ CP is the Dirac CP violation phase and β is the Majorana CP phase. There is a second Majorana phase if the lightest neutrino is not massless. As regards the CP violation, two weak basis invariants J CP [65] and I 1 [66][67][68][69][70] associated with the CP phases δ CP and β respectively can be defined,

Mixing patterns derived from S 4 with NO neutrino masses
In this section, we shall consider all possible residual subgroups arising from the breaking of S 4 flavor symmetry and CP, the resulting predictions for lepton mixing parameters and neutrino masses are studied. The group theory of S 4 and all the CG coefficients in our basis are reported in appendix A. S 4 has 20 nontrivial abelian subgroups which contain nine Z 2 subgroups, four Z 3 subgroups, three Z 4 subgroups, four K 4 ∼ = Z 2 × Z 2 subgroups. In our basis given in appendix A, the generalized CP transformation compatible with the S 4 flavor symmetry is of the same form as the flavor symmetry transformation [31], i.e.
where g can be any of the 24 group elements of S 4 . As discussed in section 2, the flavor symmetry S 4 is broken to the abelian subgroup G l which is capable of distinguishing the three generations in the charged lepton sector. Then G l can take to be any one of the 11 subgroups Z 3 , Z 4 and K 4 of S 4 . The vacuum alignments of φ atm and φ sol preserve different residual symmetries G atm H atm CP and G sol H sol CP , respectively. The residual flavor symmetries G atm and G sol can be any one of the 20 abelian subgroups of S 4 . After including residual CP symmetry, we find there are altogether 4400 kinds of possible breaking patterns. But these breaking patterns are not all independent from each other. If a pair of residual flavor symmetries {G l , G atm , G sol } is conjugated to the pair of groups {G l , G atm , G sol } under an element of S 4 , i.e., then these two breaking patterns will lead to the same predictions for mixing parameters [32,35,37]. As a result, it is sufficient to analyze the independent residual flavor symmetries not related by group conjugation and the compatible remnant CP. From appendix A, we find that all the Z 3 subgroups of S 4 are conjugate to each other, all the Z 4 subgroups are related with each under group conjugation, K is a normal subgroup of S 4 , and the other three K 4 subgroups are conjugate to each other. As a consequence, it is sufficient to only consider four types of residual symmetries in the charged lepton sector, i.e.
and K (S,U ) 4 , while both G atm and G sol can be any one of these 20 subgroups of S 4 . In the present work, we assume that the three generations of left-handed leptons doublets are assigned to transform as S 4 triplet 3. For G l being above four kinds of subgroups, up to permutations and phases of the column vectors, the The residual CP symmetries in the atmospheric neutrino sector and the solar neutrino sector have to be compatible with the residual flavor symmetries, and the restricted consistency conditions in Eqs. (2.5a) and (2.5b) must be fulfilled. For the residual flavor symmetries G atm and G sol being the 20 subgroups of S 4 , the corresponding residual CP transformations consistent with these subgroups are listed in table 1. In this work, we assume that the flavon fields φ atm and φ sol are assigned to transform as S 4 triplet 3 and 3 , respectively. In our working basis, the S 4 singlet contraction rules for 3 ⊗ 3 → 1 and As a consequence, we can read out the matrices U a and U s as follow, In other words, the column vectors v atm and v sol defined above Eq. (2.8) are v atm = P 132 φ atm /v φa and v sol = P 132 φ sol /v φs . Hence the column vectors v atm and v sol can be obtained by exchanging the second and the third elements of the columns φ atm /v φa and φ sol /v φs , respectively. The most general VEVs of the flavons φ atm and φ sol which preserve the possible residual symmetries in table 1 are summarized in table 2. For some residual flavor groups, not all the compatible residual  CP transformations in table 1 are explicitly listed in table 2, this is because the invariant vacuum  are also (0, 0, 0) T . Comparing with table 1, for some residual flavor subgroups we only show the invariant vacuum alignments for part of the CP transformations consistent with them. The reason is that the invariant VEVs for the remaining compatible CP transformations can be obtain by multiplying an overall factor i from the above given VEVs, and the contribution of the overall i can be be compensated by shifting the sign of the couplings xatm and x sol . alignments for the shown residual CP and the unshown ones only differ in an overall factor i. The contribution of the overall i can be absorbed into the couplings x atm and x sol . Following the procedures presented in section 2, we can straightforwardly obtain the expressions of the mixing parameters (three mixing angles, one Dirac CP phase and one Majorana CP phase) and the neutrino masses for each possible residual symmetry {G l , G atm H atm CP , G sol H sol CP }. In order to single out all independent viable breaking patterns from all possible breaking patterns in tri-direct CP approach. We will first find all possible independent pairs of {G l , G atm H atm CP , G sol H sol CP } which are not related by group conjugation given in Eq. (3.2). In order to quantitatively assess how well a residual symmetry can describe the experimental data on mixing parameters and neutrino masses [1], we define a χ 2 function to estimate the goodness-of-fit of a chosen values of the input parameters, where the input parameters m a , r = m s /m a and η are defined in Eq. (2.8), the parameter x parameterizes the vacuum of the flavon φ sol , O i denote the global best fit values of the observable quantities including the mixing angles sin 2 θ ij and the mass splittings ∆m 2 21 and ∆m 2 3l (∆m 2 3l = ∆m 2 31 for NO and ∆m 2 3l = ∆m 2 32 for IO), and σ i refer to the 1σ deviations of the corresponding quantities. The values of O i and σ i are taken from the global data analysis [1]. P i ∈ {sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , ∆m 2 21 , ∆m 2 3l } are the theoretical predictions for the five physical observable quantities as functions of x, η, m a , r. Here the contribution of the Dirac phase δ CP is not included in the χ 2 function. The reason is that the value of δ CP is less constrained at present. For each set of the input parameters x, η, m a and r, we can extract the predictions for P i and the corresponding χ 2 . We have carried out the χ 2 minimization. After performing the χ 2 analysis for all possible breaking patterns in tri-direct CP approach, we find eight independent interesting mixing patterns with NO and eighteen interesting mixing patterns with IO . All the viable cases  and the corresponding predictions for mixing parameters and neutrino masses are summarized in  table 3. Then we proceed to study the eight NO viable cases (five cases in this section and three cases in appendix B) one by one.
For this breaking pattern, the charged lepton mass matrix m † l m l is diagonal such that the unitary transformation U l is the identity matrix, as shown in Eq. (3.3). From table 1, we find that there are 4 possible residual CP transformations which are compatible with the residual family symmetry Z U 2 in the atmospheric neutrino sector. For the residual CP transformations X atm = {1, U }, the VEV alignment of flavon φ atm is where v φa is a real parameter with dimension of mass. For other two residual CP transformations X atm = {S, SU }, the alignment of flavon φ atm is which differs from the vacuum configuration of Eq. (3.6) by an overall factor i. We see from Eq. (2.8) that this overall factor i can be absorbed into the sign of the coupling constant x atm . Hence the two alignments in Eq. (3.6) and Eq. (3.7) will give rise to the same light neutrino mass matrix, and it is sufficient to consider one of them. For other residual symmetries discussed in the following, if two alignments of φ atm differ in an overall i only one of them would be studied as well. Without loss of generality, here we shall choose the atmospheric vacuum in Eq. (3.6), i.e. the residual CP is X atm = {1, U } in the atmospheric neutrino sector.
Firstly we consider the solar residual CP transformations X sol = {1, SU }, then the VEV of flavon field φ sol reads as where x is a dimensionless real number and v φs is a real parameter with dimension of mass. Consequently the Dirac neutrino mass matrix m D and the heavy right-handed neutrino Majorana mass matrix m N take the following form where the couplings y atm and y sol are real since the theory is invariant under CP. Using the seesaw formula, we can obtain the low energy effective light neutrino mass matrix NO for x, η, m a and r ≡ m s /m a being free parameters (G l , G atm , G sol ) X sol χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 δ CP /π β/π m 2 (meV) m 3 (meV) m ee (meV) IO for x, η, m a and r ≡ m s /m a being free parameters (G l , G atm , G sol ) X sol χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 δ CP /π β/π m 1 (meV) m 2 (meV) m ee (meV)  where an overall unphysical phase has been omitted and it would be neglected hereinafter for the other cases, and the parameters m a , m s and η are defined in Eq. (2.8). We find that the above neutrino mass matrix m ν fulfills It implies that the column vector (2, −1, −1) T is an eigenvector of m ν with zero eigenvalue. Subsequently we follow the procedure given in section 2 to perform a unitary transformation U ν1 , with where U T B is well-known tri-bimaximal (TB) mixing matrix. The neutrino mass matrix m ν is block diagonal, and its entries are given by (3.14) Furthermore, m ν be diagonalized by the unitary matrix U ν2 given in Eq. (2.16), i.e.

(3.22)
It is easy to check that θ 23 = π/4 leads to cos δ CP = 0 which corresponds to maximal CP violation δ CP = ±π/2. The neutrino masses m 2 and m 3 depend on all the four input parameters x, η, m a and m s while mixing parameters and mass ratio m 2 2 /m 2 3 only depend on x, η and r ≡ m s /m a . In the case η, m a and r being free parameters, we find that the experimental data on the mixing angles and the neutrino masses can be achieved for some special x.
In order to show concrete examples, some benchmark values of the parameter x and η are considered and the numerical results of the mixing parameters and neutrino masses are listed in table 4. The solar flavon alignment φ sol for these representative values of x takes a relatively simple form, consequently we expect it should be not difficult to be realized dynamically in an explicit model. We show the predictions for the effective Majorana mass m ee in neutrinoless double beta decay in the last column of table 4, where the effective mass m ee is defined as [64], From table 4, we can see that the measured values of the lepton mixing angles and the mass splittings ∆m 2 21 and ∆m 2 31 can be accommodated for certain choices of x, η, m a and r. For the benchmark value x = −1, the solar flavon alignment is φ sol = (1, −1, 3) T v φs , it is exactly the Littlest seesaw model with CSD(3) which is originally proposed in [15]. The solar vacuum φ sol = (1, −3, 1) T v φs for x = 3 corresponds to another version of Littlest seesaw [16]. Moreover, the value x = 4 leads to the vacuum φ sol = (1, 4, −2) T v φs , the CSD(4) scenario [17] is reproduced. From table 4, we see that a smaller χ 2 than the original LS model [15][16][17]71] can be achieved for the values of x = −1/2 and η = ±π/2, the corresponding vacuum alignment φ sol ∝ (2, −1, 5) seems simple and it should be easy to realize in a concrete model. Furthermore, we perform a comprehensive numerical analysis. The three input parameters x, r and η are randomly scanned over x ∈ [−20, 20], r ∈ [0, 20] and η ∈ [−π, π]. We only keep the points for which the resulting mixing angles sin 2 θ ij and the mass ratio m 2 2 /m 2 3 are in the experimentally preferred 3σ regions [1]. The parameter m a can be fixed by requiring that the individual squared mass differences ∆m 2 21 and ∆m 2 31 are reproduced. Then the predictions for the CP violating phase δ CP and β and the neutrino masses as well as m ee can be extracted. In the end we find the allowed regions of the parameter x is [−2.072, −0.287] ∪ [2.463, 4.683], of the parameter |η| is [0.414π, 0.861π] and of the parameter r is [0.0400, 0.166]. As regards the predictions for the mixing angles, we find that any values of sin 2 θ 13 and sin 2 θ 23 in their 3σ ranges can be achieved. The the solar mixing angle is in a narrow region 0.317 ≤ sin 2 θ 12 ≤ 0.319, which arise from the TM1 sum rule in Eq. (3.19). The predicted ranges of Dirac CP phase |δ CP | and Majorana CP phase |β| are [0.299π, 0.624π] and [0.273π, 0.608π], respectively. These predictions may be tested at future long baseline experiments, as discussed in [71]. The allowed ranges of the mixing parameters for other breaking patterns are also obtained by randomly varying the parameters x, r and η in the ranges x ∈ [−20, 20], r ∈ [0, 20] and η ∈ [−π, π]. We will not explicitly mention this point in the following.
• X sol = {S, U } From table 2, we find that the VEV of φ sol is proportional to (1, 1 + ix, 1 − ix) T . Inserting the vacuum configuration of the flavons φ atm and φ sol into Eq. (2.8), we obtain the light effective Majorana neutrino mass matrix is It is easy to check that the column vector (2, −1, −1) T is an eigenvector of m ν with zero eigenvalue. In order to diagonalize light neutrino mass matrix m ν in above equation, we first perform a unitary transformation U ν1 , where U ν1 is taken to be TB mixing matrix U T B . Then the neutrino mass matrix m ν is of block diagonal form with nonzero elements y, z and w, The neutrino mass matrix m ν can be exactly diagonalized by U ν2 shown in Eq. (2.16). It is easy to check that the PMNS matrix takes the same form as Eq. (3.16), and it is also a TM1 mixing matrix. Therefore the expressions of mixing angles and CP invariants are still given by Eqs. (3.18) and (3.21), respectively. However, the explicit dependence of the parameters y, z and w on m a , m s , η, x differs from that of the above case with X sol = {1, SU }. Hence distinct predictions for mixing parameters are reached. We can check that the neutrino mass matrix m ν in Eq. (3.24) has the following symmetry properties The former implies that the reactor and solar mixing angles are invariant, the atmospheric angle changes from θ 23 to π/2 − θ 23 and the Dirac phase changes from δ CP to π + δ CP under the transformation x → −x. The later implies that all the lepton mixing angles are kept intact and the signs of all CP violation phases are reversed by changing x → −x and η → −η. Once the φ sol /v φs x η ma(meV) r χ 2 min sin 2 θ13 sin 2 θ12 sin 2 θ23 δCP /π β/π m2(meV) m3(meV) mee(meV)  values of x and η are fixed, the light neutrino mass matrix m ν would only depend on two free parameters m a and m s whose values can be determined by the neutrino mass squared differences ∆m 2 21 and ∆m 2 31 . Then we can extract the predictions for three lepton mixing angles and CP violation phases δ CP and β. The best fit values of the mixing parameters and neutrino masses for some benchmark values of x and η are shown in table 5. The most interesting points are η = 0 and π which predict maximal atmospheric mixing angle, maximal Dirac phase and trivial Majorana phase. The reason is because the general neutrino mass m ν shown in Eq. (3.24) has an accidental µτ reflection symmetry in the case of η = 0 and π [72]. The realistic values of mixing angles and mass ratio m 2 2 /m 2 3 can be obtained for x = ±4, ±7/2, ±7/6 in the case of η = 0 or π. In order to describe the experimental data at 3σ level [1], the three input parameters are constrained to be |x| ∈ The unitary transformation U l is an identity matrix up to permutation of columns because the residual symmetry G l = Z T 3 is diagonal in our working basis. The possible residual CP transformations X sol and the corresponding VEVs of the flavon field φ sol are the same as those of case N 1 .
where v φa is a real number with dimension of mass. For X sol = {1, SU }, the alignment of φ sol is given in Eq. (3.8). Then the light neutrino mass matrix is It can be block diagonalized by the TB mixing matrix, Eq. (3.29) implies that the first column of U T B is an eigenvector of m ν with zero eigenvalue. Hence the lepton mixing matrix is the TM1 mixing pattern. In order to achieve the Dirac CP phase δ CP around −π/2 which is preferred by the present data [1], we take U l = P 132 . Then the PMNS matrix can be obtained by exchanging the second and third rows of the mixing matrix in Eq. . We see that this breaking pattern can accommodate a nearly maximal Dirac CP phase. The other mixing angles except θ 12 can take any values within their 3σ ranges. The solar mixing angle sin 2 θ 12 is close to 0.318 and this is generally true for TM1 mixing, as shown in Eqs. (3.19, 3.20).
The explicit form of the vacuum of φ atm and φ sol invariant under the assumed residual symmetries can be found from table 2, i.e. φ atm ∝ 1, −2ω 2 , −2ω T and φ sol ∝ (1, 1 + ix, 1 − ix) T . The most general neutrino mass matrix takes the following form We can perform a TB transformation to obtain the block diagonal neutrino mass matrix m ν . The non-vanishing elements y, z and w of m ν are given by We can further introduce the unitary transformation U ν2 to diagonalize the neutrino mass matrix m ν , as generally shown in Eqs. (2.15,2.16). As a consequence, the lepton mixing matrix is also the TM1 pattern, and the sum rules in Eq. (3.19) and Eq. (3.22) are satisfied as well. However, the dependence of the mixing parameters on the input parameters m a , m s , η and x are different, φ sol /v φs x η ma(meV) r χ 2 min sin 2 θ13 sin 2 θ12 sin 2 θ23 δCP /π β/π m2(meV) m3(meV) mee(meV) Here we choose many benchmark values for the parameters x and η. Notice that the lightest neutrino mass is vanishing m1 = 0.
consequently the above two mixing patterns of N 2 with X sol = {1, SU } and X sol = {S, U } lead to different predictions. In table 7, we present the results of our χ 2 analysis for some simple values of x and η. We find that accordance with experimental data can be achieved for certain values of m a and r. In the case of η = π, both the atmospheric mixing θ 23 and the Dirac CP phase δ CP are maximal, while the Majorana CP phase β is trivial. We notice that realistic values of mixing angles and m 2 2 /m 2 3 can be obtained for x = 3/2, −3 in the case of η = 0 or π. If requiring sin 2 θ ij and m 2 2 /m 2 3 lie in their 3σ regions [1], we find the allowed regions of the parameter x is [−12.192 in the solar neutrino sector, the residual CP transformation X sol can only be X sol = {S, U } in order to achieve agreement with experimental data. In this case the charged lepton diagonalization matrix U l is also the identity matrix, and the vacuum expectation values of flavon fields φ atm and φ sol read as
The most general neutrino mass matrix is determined to be We find that the column vector (2, −1, −1) T is an eigenvector of m ν with zero eigenvalue. This neutrino mass matrix m ν can be simplified into a block diagonal form by performing a U T B transformation. Then we can obtain the three nonzero elements y, z and w: The neutrino mass matrix m ν can be diagonalized by performing the unitary transformation U ν2 . Thus the lepton mixing matrix is the TM1 pattern shown in Eq. Furthermore, we find that the neutrino mass matrix m ν in Eq. (3.31) has the following symmetry properties These identities indicate that the mixing angles θ 12 and θ 13 keep invariant, θ 23 becomes π/2 − θ 23 and the Dirac phase changes from δ CP to π + δ CP under the transformation x → −x. Moreover, by changing x to −x and η to −η simultaneously, all the lepton mixing angles are unchanged and the signs of all CP violation phases are reversed. Detailed numerical analyses show that accordance with experimental data can be achieved for certain values of x, m a , r and η, and the corresponding benchmark numerical results are listed in table 8. We find that acceptable values of mixing angles and m 2 2 /m 2 3 can be obtained for x = ±4, η = π and x = ±11/2, η = 0. If all the three mixing angles and m 2 2 /m 2 3 are restricted to their 3σ regions [1]. The viable ranges of the input parameters |x| and r are [3.641, 5.911] and [0.213, 0.568] respectively while any value of η ∈ [−π, π] is viable. Then the atmospheric mixing angle sin 2 θ 23 and the Dirac CP phase δ CP are predicted to be 0.458 ≤ sin 2 θ 23 ≤ 0.542 and |δ CP | ∈ [0.443π, 0.557π], respectively. The Majorana CP phase β can take any value between −π and π.
In short summary, We find that all the above three breaking patterns N 1 , N 2 and N 3 predict TM1 lepton mixing matrix and the experimental data [1] can be described very well. All the three breaking patterns predict a normal mass hierarchy with m 1 = 0 and the sum rules in Eqs. (3.19) and (3.22). In fact these two sum rules are common to all TM1 mixing matrices. The prospects for testing the two sum rules in future neutrino facilities have been discussed [73].
Under the assumption of TM1 mixing, the structure of the Dirac mass matrix has been analyzed in Refs. [74,75] in the framework of two right-handed neutrino seesaw model, generally more parameters are involved than the tri-driect CP models.
, U } This breaking pattern has been studied in great detail by us [58]. Hence we shall not repeat the analysis here. When all the three lepton mixing angles and the neutrino mass ratio m 2 2 /m 2 3 are restricted in their 3σ regions [1], we find that the parameters x, |η| and r should be in the ranges of [−6.238, −3.365], [0.347π, π] and [0.154, 0.607], respectively. Moreover, we show the results of χ 2 analysis for some benchmark values of x and η in table 9. The highlighted case with red background in table 9 has been realized in a concrete model [58]. In section 5 of the present work, we shall construct a model to realize the breaking pattern with x = −4 and η = ±3π/4. The corresponding best fit values of the input parameters, mixing angles, CP phases and neutrino masses are highlighted with green background in table 9.
T } In this case, the unitary transformation U l is TB the mixing matrix U T B . From table 2, we find that the vacuum alignments of the flavons φ atm ∼ 3 and φ sol ∼ 3 are dictated by the residual x η m a (meV) r χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 δ CP /π β/π m 2 (meV) m 3 (meV) m ee (meV) Here we choose many benchmark values for the parameters x and η. Notice that the lightest neutrino mass is vanishing m1 = 0. symmetry to be It is easy to check that the two column vectors φ atm and φ sol are orthogonal to each other, i.e. φ atm † φ sol = 0. This scenario is referred to as form dominance in the literature [59][60][61]. Applying the seesaw relation in Eq. (2.8) gives rise to a light neutrino mass matrix, (3.38) In this case, φ atm and φ sol are proportional to two columns of U ν which is the diagonalization matrix of m ν , It implies that the three neutrino masses are 0, 2m a and 1 + 2x 2 m s which are independent of the phase η. Including the contribution U l = U T B from the charged lepton sector, we find the lepton mixing matrix is given by The three lepton mixing angles read as , , which are expressed in terms of one real parameter x. Furthermore, we can derive the following sum rules among the mixing angles sin 2 θ 12 cos 2 θ 13 = 1 4 , sin 2 θ 23 = 6 − 7 sin 2 θ 13 ± 2 sin θ 13 2(3 − 4 sin 2 θ 13 ) 9 cos 2 θ 13 6 − sin 2 θ 13 ± 2 √ 6 sin θ 13 9 , (3.43) where the first sum rule in Eq. (3.43) has already appeared in the literature [37], and the sign "±" in the second sum rule depends on the value of x. For the best fitting value of the reactor angle sin 2 θ 13 = 0.02241 [1], the solar mixing angle is determined to be sin 2 θ 12 = 0.256 and the atmospheric mixing angle is sin 2 θ 23 = 0.746 or sin 2 θ 23 = 0.582. We see that the later value of sin 2 θ 23 is compatible with the preferred values from global data analysis [1]. The value of sin 2 θ 12 is rather close to its 3σ lower limit 0.275. As a result, we suggest this mixing pattern is a good leading order approximation since accordance with experimental data should be easily achieved after subleading contributions are taken into account in a concrete model. Furthermore, we find that the two CP rephasing invariants J CP and I 1 are Hence the Dirac CP phase is trivial for any value of x. From the expressions of mixing angles in Eq. (3.42) and Majorana invariant in Eq. (3.44), we find that the Majorana CP phase β is determined to be It implies that trivial Majorana CP phase is obtained for η = 0 or π. For η = ±π/2, the Majorana CP phase is maximal. As an example, we take the representative value x = −5/4. Thus the VEV of the flavon field φ sol is proportional to 1, − 5 4 ω, 5 4 ω 2 T and the PMNS matrix is Wee see that the reactor and atmospheric mixing angles are compatible with the preferred values from global fit [1] at the 3σ level. Furthermore, in the case of x = −5/4 the two neutrino mass squared differences only depend on the values of m a and r, as shown in figure 2. We find that the best fit values of the two neutrino mass squared differences ∆m 2 21 and ∆m 2 32 can be reproduced. In order to increase the readability of the paper, the remaining three viable cases N 6 , N 7 and N 8 are moved to the appendix B. The reason is that the diagonalization matrix of the charged lepton mass matrix and the phenomenologically interesting alignments of the flavon φ sol may be not simple enough to be realized in a concrete model. to study five viable cases among them and present their predictions for lepton mixing angles, CP violating phases and neutrino masses. The other viable breaking patterns are shown in appendix C. The last two of the five breaking patterns in this section will lead to the form dominance texture.
The diagonalization matrix of the charged lepton mass matrix m † l m l is an unity matrix because of the residual symmetry G l = Z T 3 . The given residual symmetries fix the vacuum of the flavon fields φ atm and φ sol to be Thus the light neutrino mass matrix is given by It can be simplified into a block diagonal form m ν by performing a unitary transformation U ν1 , where the unitary matrix U ν1 is As shown in Eq. (2.13), m ν can be parametrized by three parameters y, z and w with The neutrino mass matrix m ν can be exactly diagonalized by a second unitary transformation U ν2 in Eq. (2.16). Then the lightest neutrino mass m 3 is vanishing and the other two neutrino masses m 1,2 can be obtained from Eq. (2.17). The lepton mixing matrix is determined to be In the IO case, P ν will always be defined as Eq. (4.6) and it will be omitted for simplicity in the following. From the lepton mixing given in Eq. (4.5), we can extract the expressions of mixing angles and CP invariants as follow, where the definitions of CP invariants is given in Eq. (2.23). Note that the atmospheric mixing angle θ 23 is maximal for any combination of input parameters and the reactor mixing angle θ 13 only depend on input parameter x. In order to show concrete examples, we give the predictions for mixing parameters, neutrino masses and the effective mass in neutrinoless double beta decay for x = −1/8 which gives a relatively simple VEV of φ sol . In the case of x = −1/8, we find the third column of the PMNS matrix is  [1]. In order to evaluate how well the predicted mixing patterns agree with the experimental data on mixing angles and neutrino masses, we shall perform a χ 2 analysis which uses the global fit results of Ref. [1]. The χ 2 analysis results are η = −0.983π, m a = 5.721 meV, r = 8.673, χ 2 min = 20.595, sin 2 θ 13 = 0.0215 , Here only the residual CP symmetry X sol = {U, T } can give phenomenologically viable mixing patterns. The VEV of φ sol is φ sol = v φs 1, xω, xω 2 T . From the general VEVs of flavons φ atm and φ sol , we find the general form of the neutrino mass matrix is given by After perform a unitary transformation U ν1 to m ν , where unitary matrix U ν1 takes the following form Then neutrino mass matrix m ν is a block diagonal matrix with the nonzero elements y, z and w being The general form of the diagonalization matrix of m ν is U ν2 which given in Eq. (2.16) for IO case.
As the diagonalization matrix of the charged lepton mass matrix is the identity matrix. Then the PMNS matrix reads The lepton mixing parameters are predicted to be Note that the atmospheric mixing angle θ 23 is predicted to be 45 • . The admissible range of x is [−0.153, −0.138] ∪ [0.108, 0.118] which is obtained from the requirement that θ 13 is in the experimentally preferred 3σ range. In order to see how well the lepton mixing angles and the neutrino masses can be described by this breaking pattern and its prediction for CP phases, we perform a χ 2 analysis for x = −1/7 and η = π. From PMNS matrix in Eq. (4.12), we see that the fixed column of PMNS matrix is We see that the results of η = π are viable and it leads to maximal Dirac CP phase and trivial Majorana CP phase. The reason is that the neutrino mass matrix in Eq. (4.9) has the symmetry property m ν (x, r, η) = P T 132 m * ν (x, r, η)P 132 . The reason why we chose η = π is that it is easy to realize in an explicit model, please see the model in Ref. [58]. Furthermore, we perform a comprehensive numerical analysis. When the 3σ constraints on mixing angles and mass ratio m 2 1 /m 2 2 are imposed. We find that the allowed regions of the parameter The mixing angles θ 13 and θ 12 can take any values in their 3σ ranges.
In this combination of residual flavor symmetries, both the residual CP transformations X sol = {1, U } and X sol = {U, S} will give results which agree with the experimental data. For the former case, the general VEVs invariant under the actions of the residual symmetries in the atmospheric neutrino and the solar neutrino sectors are The light neutrino mass matrix is given by the seesaw formula, yielding The diagonalization matrix of above neutrino mass matrix can be taken to be where Form the expression of U ν , we find that the lepton mixing matrix takes the following form Accordingly we find the expressions of the mixing angles and CP invariants are  We see that θ 13 is rather close to its 3σ lower limit 0.2068 [1]. Hence this results should be considered as a good leading order approximation. The reason is that accordance with experimental data can be easily achieved after subleading contributions are taken into account.
For this kind of residual CP symmetries in the atmospheric neutrino sector and the solar neutrino sector, the neutrino mass matrix takes the form Then we find that the neutrino mass matrix m ν is a block diagonal matrix with nonzero elements y = 3m a − 3x 2 m s e iη , z = −3ix 1 + x 2 m s e iη , w = 3 1 + x 2 m s e iη . (4.25) The neutrino mass matrix m ν can be further diagonalized by the unitary matrix U ν2 . Then the lepton mixing matrix is determined to be of the form The lepton mixing angles and CP invariants turn out to take the form The atmospheric mixing angle is maximal and the value of θ 13 relies on the value of x. If θ 13 is required in its 3σ range, we find that |x| has to lie in the range of , Z T U 2 , Z T U 2 ), X atm = {U, T }, X sol = {U, T } In this case, only the residual CP transformation X sol = {U, T } will give results which agree with the experimental data. This mixing pattern is discussed in the case N 5 with NO. Then the results of this case can be easy obtained from the predictions of case N 5 . The PMNS matrix in this case is One can straightforwardly extract the lepton mixing angles as follows, The three mixing angles only depend on one input parameter x. Then two sum rules among the three mixing angles can be obtained. We find that the two sum rules here are the same as the sum rules in Eq. (3.43). As same as N 5 , the two CP phases are predicted to be sin δ CP = 0, β = η + π . (4.31) It implies that the Dirac CP phase is trivial for input parameters taking any values. The Majorana CP phase relies only on input parameter η. In the case η = 0 or π, the Majorana CP phase is also trivial. While the Majorana CP phase is maximal for η = ±π/2. As an example, we give the predictions for x = 4. Here the VEVs of the flavonr fields φ sol is proportional to column vector 1, 4ω, 4ω 2 T and the PMNS matrix is We find that the solar mixing angles is rather close to its 3σ lower bound [1]. Hence this mixing pattern can be regarded as a good leading order approximation.
It is easy to check that the column vectors φ atm and φ sol are orthogonal to each other. It leads to a form dominance breaking pattern [59][60][61]. The light neutrino mass matrix is given by the seesaw relation with The diagonalization matrix of m ν with non-negative eigenvalues can be taken to be where Only input parameters m a , m s and x are involved in the two nonzero neutrino masses. Since the diagonalization matrix of the charged lepton mass matrix is a 3 × 3 identity matrix up to the permutation of columns. Then lepton mixing matrix can be taken to be U ν , i.e. this form dominance breaking pattern predicts a TM1 mixing matrix. Then the three lepton mixing angles and two CP invariants are determined to be sin 2 θ 13 = 2x 2 9 + 6x 2 , sin 2 θ 12 = 3 9 + 4x 2 , sin 2 θ 23 = 1 2 , We note that the atmospheric mixing angle θ 23 is maximal and the other two mixing angles depend on a single real parameter x. Hence the following sum rule between the reactor mixing angle and the solar mixing angle is found to be satisfied cos 2 θ 12 cos 2 θ 13 = 2 3 . (4.39) This sum rule comes from the so called TM1 mixing matrix. From the CP invariants defined in Eq. (2.23) and the expressions of mixing parameters in (4.38), we find It predicts a maximal Dirac CP phase and the Majorana CP phase β equals the negative value of input parameter η. As an example, we give the predictions for x = 1/3. Here the VEVs of the flavon φ sol is proportional to column vector 1, 1 + i 3 , 1 − i 3 T and the PMNS matrix for The three mixing angles are predicted to be These results agrees with all measurements to date [1]. In above we have given five kinds of viable breaking patterns with IO. From table 3, we know that other thirteen kinds of breaking patterns with IO are left. In order to increase an article's readability, we shall discuss them in appendix C.

Model construction
In section 3, section 4, appendix B and appendix C, we have performed a model-independent analysis for the lepton mixing patterns which can be derived from S 4 H CP in the tri-direct CP approach. In the model-independent analysis, we have assumed that the flavons φ atm and φ sol transform as S 4 triplet 3 and 3 , respectively. From model-independent analysis, we find 8 kinds of breaking patterns which are compatible with current experimental data for NO neutrino masses. In this section, we shall construct a supersymmetric model with the flavor symmetry S 4 and a CP symmetry, and the symmetry breaking pattern N 4 is realized due to on-vanishing vacuum expectation values of some flavons. The phenomenological predictions of N 4 for lepton mixing parameters and neutrino masses has been studied in detail in Ref. [58]. The reason why we construct a model to realize the breaking pattern N 4 are: firstly the TM1 mixing matrix obtained in N 1 ∼ N 3 has been widely discussed. Secondly the vacuum alignment of flavon φ sol and the fixed column of PMNS matrix are relatively simpler than other breaking patterns which don't give TM1 mixing. Thirdly the minimum χ 2 in N 4 is small for simple benchmark values of x and η. We have tabulated many simple admissible values of x and η in table 9. We find that phenomenologically viable lepton mixing matrix can be obtained for the atmospheric flavon vacuum alignment (1, ω 2 , ω) T and the solar flavon vacuum alignment (1, −4, −4) T with η = ± 3π 4 . In the present work, the model based on above vacuum alignments will be realized, i.e. we shall present a supersymmetric model which contains two right-handed neutrinos and realizes the breaking pattern N 4 with x = −4 and η = ± 3π 4 . Here we assume that the three generations of left-handed lepton doublets L furnish triplet representation 3 under the family symmetry S 4 , while the right-handed charged leptons e c , µ c and τ c are singlet 1 under the family symmetry S 4 . The two right-handed neutrinos ν c atm and ν c sol are assumed to be the singlet representations 1 and 1 , respectively. In order to ensure the needed vacuum alignment and to forbid unwanted couplings, the auxiliary symmetry Z 5 × Z 8 × Z 8 is imposed. In this model, the original symmetry S 4 H CP will spontaneous break to Z T 3 , Z T ST 2 2 × X atm and Z U 2 × X sol in the charged lepton, atmospheric neutrino and solar neutrino sectors, where the residual CP transformations X atm = SU and X sol = U . As a consequence, the desired vacua φ atm ∝ 1, L e c µ c τ c ν c atm ν c sol H u,d η l φ l ξ a φ a ξ s η s χ s ϕ s ∆ s φ s ψ s ξ 0 l φ 0 l φ 0 a σ 0 ρ 0 η 0 χ 0 ϕ 0 ∆ 0 κ 0 S 4 3 1 1 1 1 1 1 2 3 1 3 1 2 3 3 3 3 3 1 3 3 2 2 2 3 3 3 1 Z 5 ω 4 5 ω 3 5 ω 4 5 1 1 1 1 ω 5 ω 5 1 ω 5 1 ω 3 5 ω 2 5 1 ω 3 5 ω 5 ω 4 5 ω 3 5 ω 3 5 ω 3 5 ω 2 5 ω 5 ω

Vacuum alignment
We adopt the now-standard F −term alignment mechanism to generate the appropriate vacuum alignments of the flavor symmetry breaking flavons. The leading order (LO) driving superpotential w d which is invariant under the imposed S 4 × Z 5 × Z 8 × Z 8 takes the following form where w l d , w atm d and w sol d are used to realize the LO vacuum alignments of the flavons in the charged lepton sector, the atmospheric neutrino sector and the solar neutrino sector, respectively. They can be expressed as where the subscript of (· · · ) r denotes a contraction of the S 4 indices into the representation r. All the coupling g i h 1 , f i and mass parameters M ϕ , M ∆ in Eq. (5.2) are real. The reason is that we have required that the theory is invariant under the imposed generalised CP symmetry. In the SUSY limit, the vacuum alignment is achieved via the requirement of vanishing F −terms of the driving fields. In the charged lepton sector, the F −term conditions obtained from the driving fields ξ 0 l and φ 0 l are given by By straightforward calculations, we find that these equations are satisfied by the alignment where v η l is undetermined. In the atmospheric neutrino sector, the vacuum is determined by F −term conditions associated with the driving field φ 0 a ∂w atm d ∂φ 0 These equations lead to the vacuum alignment of φ a being Now we turn to the F −term conditions of the solar neutrino sector. The resulting F -term conditions depend on the S 4 representation of ρ 0 and σ 0 are given as The above equations lead to the vacuum alignments of χ s and ψ s as follow The alignment of the doublet flavon η originates in the F -term A straightforward calculation shows that the F -term conditions resulting from above determine the doublet alignment uniquely to From these results, we immediately see the F -term conditions generate the alignment Similarly, we can consider the F -term of the driving field ϕ 0 ∂w sol These conditions determine the alignments of ∆ s being (5.14) In order to realise the alignment φ s ∝ (1, −4, −4) T , we consider the F -term of driving field ∆ 0 : Now we have obtained the vacuum alignments of all flavons η s , ψ s , χ s , ϕ s , ∆ s and φ s in the solar neutrino sector by adopting the standard F −term alignment mechanism. In other words, the needed vacuum alignment φ s ∝ (1, −4, −4) T is realized. Next we shall fix the overall phases of all VEVs of flavons in the atmospheric and solar neutrino sectors. From the alignments of flavons ξ s , η s , χ s , ψ s , ϕ s and φ s shown above, we find the VEVs of these fields are invariant under the subgroup Z U 2 . In order to obtain the phase with η = ± 3π 4 , we introduce the S 4 singlet fields in table 11. Then the driving superpotential which is used to obtain the phases of all the VEVs of flavons in the neutrino sector is   where couplings x i and mass parameters M 2 i and M Ω i are all real. The F-term conditions from above superpotential are ∂w phase In order to obtain the observed hierarchy among the charged lepton masses, we assume where λ is the Cabibbo angle with λ 0.23. Moreover, the VEVs of flavons in the neutrino sector are expected to be of the same order of magnitude and we will take them to be of the same order as the VEVs of flavons in the charged lepton sector, i.e.
where v Ω i (i = 1, · · · , 6) are the VEVs of flavons Ω i . Now we will briefly touch on the subleading corrections to the driving superpotential given above. We first start with the corrections to the driving superpotential w l d which contains the driving fields ξ 0 l and φ 0 l . We find that the NLO corrections of it is suppressed by 1/Λ 2 with respect to the renormalizable terms in Eq. (5.2). The subleading contributions to driving superpotential w atm d and w sol d involve three flavon fields. The corresponding corrections to the leading order terms in w atm d and w sol d are of relative order λ 2 .

The structure of the model
The lowest dimensional Yukawa operators of the charged lepton mass terms, which are invariant under the imposed flavor symmetry S 4 × Z 5 × Z 8 × Z 8 , can be written as where all couplings are real due to the generalized CP symmetry. Substituting the flavon VEVs of η l and φ l in Eq. (5.4), we find the charged lepton mass matrix is diagonal with the three charged lepton masses being where v d = H d . Note, in order to obtain the mass hierarchies of the charged leptons m e : m µ : m τ λ 4 : λ 2 : 1, the auxiliary symmetry Z 8 is imposed, where λ 0.23 is the Cabibbo angle. The auxiliary symmetry Z 8 imposes different powers of η l and φ l to couple with the electron, muon and tau lepton mass terms. From Eq. (5.25), we find that the electron, muon and tau masses arise at order ( Φ l /Λ) 3 , ( Φ l /Λ) 2 and Φ l /Λ respectively, where Φ l refer to either η l or φ l . If we assume that Φ l /Λ is of order λ 2 , then the mass hierarchy of the charged leptons can be reproduced. Moreover, the subleading operators related to e c , µ c and τ c comprise four flavons and consequently are suppressed by 1/Λ 4 . Such corrections for the charged lepton masses and lepton mixing parameters can be neglected. Now we come to the neutrino sector. The light neutrino masses are given by the famous type-I seesaw mechanism with two right-handed neutrinos. The most general LO superpotential for the neutrino masses is where the four coupling constants y a , y s , x a and x s are real because the theory is required to be invariant under the generalised CP transformation. From the vacuum alignments of flavons φ a and φ s , we can read out the Dirac and Majorana mass matrices as follows where v u = H u and the clarity of expressions of VEVs v ξa , v ξs , v φa , v φs are shown in section 5.1.
After applying the seesaw formula, the effective light neutrino mass matrix can be written as In section 5.1, we have taken the solution with the phase of the ratio v 2 φs v ξa v 2 φa v ξs being ± 3π 4 . Up to the over all phase of the neutrino mass matrix in Eq. (5.29), we see that this neutrino mass matrix is of the same form as the general mass matrix in breaking pattern N 4 [58] but with It is easy check that we can obtain the results with η = ± 3π 4 in the case of x a x s > 0. In the following, we will briefly touch on the subleading corrections to the superpotential given in sections 5.1 and 5.2. Furthermore,we find that the next-to-leading operators of w ν are suppressed by 1/Λ 2 with respect to the LO contributions and therefore can be negligible.
From the standard procedure shown in section 2, we find that the above model predicts the following LO lepton mixing matrix where the diagonal phase P ν is given in Eq. (3.17). All the parameters θ, ψ, σ and ρ only depend on one input parameter r. In the case of η = − 3π 4 , the three mixing angles and the two CP invariants can be expressed in terms of r = m s /m a as where parameter C r is defined as The neutrino masses m 2 2 and m 2 3 are dependent on free parameters m a and r. If we require that all the three lepton mixing angles and two mass squared differences lie in their corresponding experimentally 3σ intervals [1]. Then the lepton mixing parameters and the neutrino masses are predicted to be 0.3362 ≤ sin 2 θ 12 ≤ 0.3364, 0.02254 ≤ sin 2 θ 13 ≤ 0.02280, 0.556 ≤ sin 2 θ 23 ≤ 0.564, −0.418 ≤ δ CP /π ≤ −0.406, 0.263 ≤ β/π ≤ 0.264, 2.690 meV ≤ m ee ≤ 2.985 meV, 8.240 meV ≤ m 2 ≤ 8.950 meV, 49.265 meV ≤ m 3 ≤ 51.235 meV .
(5. 37) We see that all mixing parameters and neutrino masses are restricted in rather narrow regions. It is straightforward to show that the model above is a powerful model to predicted lepton mixing parameters and neutrino masses, especially for mixing angle θ 12 . The next generation reactor neutrino oscillation experiments JUNO [76] and RENO-50 [77] expect to reduce the error of θ 12 to about 0.1 • or around 0.3%. The oscillation parameters θ 12 , θ 23 and δ CP would be precisely measured by the future long baseline experiments DUNE [78][79][80], T2HK [81], T2HKK [82]. Hence this breaking pattern can be checked by future neutrino facilities. Furthermore, we expect that a more ambitious facility such as the neutrino factory [83][84][85] could provide a more stringent tests of our approach. We see that the light neutrino mass matrix in Eq. (5.29) has the following symmetry property m ν (m a , r, −η) = P T 132 m * ν (m a , r, η)P 132 .

(5.38)
Therefore the atmospheric mixing angle changes from θ 23 to π/2 − θ 23 , the Dirac CP phase changes from δ CP to π − δ CP , the Majorana CP phase will become the opposite and other observable quantities remain unchanged under the transformation η → −η. The predictions for η = 3π 4 can be easily be obtained from the results of η = − 3π 4 . Hence we shall not show the predictions for η = 3π 4 .

Conclusion
In the present paper, guided by the principles of symmetry and minimality, we have analyzed the possible symmetry breaking patterns of S 4 H CP in the tri-direct CP approach [58] based on the two right-handed neutrino seesaw mechanism. In the tri-direct CP approach, the high energy flavor and generalized CP symmetry S 4 H CP is spontaneously broken down to an abelian subgroup G l (non Z 2 subgroups) in the charged lepton sector, to G atm H atm CP in one right-handed neutrino sector and to G sol H sol CP in other right-handed neutrino sector, as illustrated in figure 1. In this work, we assume that the flavon field φ atm which couples to right-handed N atm and the lefthanded lepton doublets L is assigned to transform as S 4 triplet 3, and the flavon φ sol which couples to right-handed N sol and the left-handed lepton doublets L transforms as the three-dimensional representation 3 under S 4 . Then the two columns of the neutrino Dirac mass matrix are determined by the vacuum alignments of φ atm and φ sol , respectively. Furthermore, we have given the basic procedure of predicting lepton flavor mixing and neutrino mass from residual symmetries in the tri-direct CP approach in a model independent way and we find that first (third) column of PMNS matrix is fixed by the diagonalization matrix U l of the charged lepton mass matrix and the vacuum alignments of φ atm and φ sol for NO (IO) spectrum.
After considering all possible breaking patterns arising from S 4 flavor symmetry combined with the corresponding generalized CP symmetry in a model independent way, we find eight phenomenologically interesting mixing patterns with NO spectrum labeled as N 1 ∼ N 8 and eighteen phenomenologically interesting mixing patterns with IO spectrum labeled as I 1 ∼ I 18 , please see table 3. For each phenomenologically interesting mixing pattern, we have analyzed the corresponding predictions for the PMNS matrix, the lepton mixing parameters, the neutrino masses and the effective mass in neutrinoless double beta decay in a model independent way in the tri-direct CP approach. There are one form dominance breaking pattern with NO spectrum (N 5 ) and two form dominance breaking patterns with IO spectrum (I 4 and I 5 ). We find that three kinds of breaking patterns with NO spectrum (N 1 ∼ N 3 ) and one form dominance breaking pattern with IO spectrum (I 5 ) yield TM1 mixing matrix. For each of these four kinds of breaking patterns with TM1 mixing matrix, two sum rules among mixing angles and Dirac CP phase corresponding to TM1 mixing are obtained. Furthermore, we perform a numerical analysis for each breaking pattern which is able to give a successful description of the lepton mixing parameters and the neutrino masses in terms of four real input parameters x, η, m a and r. In the breaking patterns with NO spectrum, we also give the χ 2 results for some benchmark values of x and η, where the parameter x comes from the VEV of flavon φ sol . The simple values of x and η are very useful in model building. Once the values of x and η are fixed, we obtain a highly predictive theory of neutrino mass and lepton mixing, in which all lepton mixing parameters and the neutrino masses are determined by only two real input parameters m a and r. In the breaking pattern N 1 , for the benchmark value x = −1 which leads to φ sol = (1, −1, 3) T v φs , it is exactly the Littlest seesaw model with CSD (3) which is originally proposed in [15]. The solar vacuum φ sol = (1, −3, 1) T v φs for x = 3, it corresponds to another version of Littlest seesaw [16]. Moreover, for the vacuum φ sol = (1, 4, −2) T v φs with x = 4, the CSD(4) scenario [17] is reproduced. Furthermore, we show the best fit values of the neutrino masses and the mixing parameters for a simple value of x for each of the eighteen breaking patterns with IO spectrum. Guided by above model independent analysis, we construct a successful flavor model involving two right-handed neutrinos based on S 4 and generalized CP symmetry to realise the breaking pattern N 4 with x = −4 and η = ± 3π 4 , in which the original symmetry S 4 H CP is spontaneously broken down to Z T 3 in the charged lepton sector, to Z T ST 2 2 ×X atm in the atmospheric neutrino sector and to Z U 2 × X sol in the solar neutrino sector, where the residual CP transformations X atm = SU and X sol = U . In this model, the first column of PMNS matrix is fixed to be 2 6 37 , 13 74 , 13 74 T . This model has not so far appeared in the literature. We find that this model is a powerful model to predicted lepton mixing parameters and neutrino masses. In particular, all the lepton mixing parameters and the neutrino masses are restricted in rather narrow regions in this model as in Eq. (5.37).
In summary, we have performed an exhaustive analysis of all possible breaking patterns arising from S 4 H CP in a new tri-direct CP approach to the minimal seesaw model with two righthanded neutrinos and have constructed a realistic flavour model along these lines. According to this approach, separate residual flavour and CP symmetries persist in the charged lepton, "atmospheric" and "solar" right-handed neutrino sectors, resulting in three symmetry sectors rather than the usual two of the semi-direct CP approach. Following the tri-direct CP approach, we have found twenty-six kinds of independent phenomenologically interesting mixing patterns. Eight of them predict a normal ordering (NO) neutrino mass spectrum and the other eighteen predict an inverted ordering (IO) neutrino mass spectrum. For each phenomenologically interesting mixing pattern, the corresponding predictions for the PMNS matrix, the lepton mixing parameters, the neutrino masses and the effective mass in neutrinoless double beta decay are given in a model independent way. One breaking pattern with NO spectrum and two breaking patterns with IO spectrum corresponds to form dominance. We have found that the lepton mixing matrices of three kinds of breaking patterns with NO spectrum and one form dominance breaking pattern with IO spectrum preserve the first column of the tri-bimaximal (TB) mixing matrix, corresponding to the TM1 mixing matrix.

Appendix
A Group Theory of S 4 S 4 is the permutation group of four objects, and it has 24 elements. In the present work, we shall adopt the same convention as [31]. The S 4 group can be generated by three generators S, T and U with the multiplication rules S 4 group has twenty abelian subgroups which contain nine Z 2 subgroups, four Z 3 subgroups, three Z 4 subgroups and four K 4 ∼ = Z 2 × Z 2 subgroups. These abelian subgroups can be expressed in terms of the generators S, T and U as follows.
• Z 2 subgroups The former six Z 2 subgroups are conjugate to each other, and the latter three subgroups are related to each other by group conjugation as well.
which are related with each other under group conjugation.
• Z 4 subgroups All the above Z 4 subgroups are conjugate to each other.
is a normal subgroup of S 4 , and the other three K 4 subgroups are conjugate to each other. S 4 has five irreducible representations which contain two singlet irreducible representations 1 and 1 , one two-dimensional representation 2 and two three-dimensional irreducible representations 3 and 3 . In this work, we choose the same basis as that of [31], i.e. the representation matrix of the generator T is diagonal. The representation matrices for the three generators are listed in table 12. Moreover, the Kronecker products of two irreducible representations of S 4 group are where R stands for any irreducible representation of S 4 . We now list the CG coefficients for our basis. All the CG coefficients can be reported in the form of R 1 ⊗ R 2 , where R 1 and R 2 are two irreducible representations of S 4 . We shall use α i to denote the elements of first representation and β i stands for the elements of the second representation of the tensor product. For the product of the singlet 1 with a doublet or a triplet, we have The CG coefficients for the products involving the doublet representation 2 are found to be Finally, for the products among the triplet representations 3 and 3 , we have Other mixing patterns with NO Here the diagonalization matrix of charged lepton mass matrix U l is given in Eq. (3.3). Here only residual CP transformations X sol = {U, S} can accommodate the present experimental data on lepton mixing. In this kind of breaking pattern the VEVs of flavon fields φ atm and φ sol are Straightforward calculations demonstrate that the neutrino mass matrix is of the following form The neutrino mass matrix m ν has an eigenvalue 0, and the corresponding eigenvector is (0, 1 + ix, −1 + ix) T . It is convenient to firstly perform a constant unitary transformation U ν1 . The unitary matrix U ν1 can take the form Then the neutrino mass matrix m ν is a block diagonal matrix and it can be diagonalized by a unitary matrix U ν2 in the (2,3) sector. The nonzero parameters y, z and w in the m ν are given by Then the lepton mixing matrix is determined to be One can straightforwardly extract the lepton mixing angles and CP phases as follows , A sum rule between the solar mixing angle θ 12 and the reactor mixing angle θ 13 is satisfied, .
For the fixed value of x, mixing angles θ 13 and θ 12 are strongly correlated with each other. On the other hand, the 3σ ranges of θ 13 and θ 12 [1] will restrict the possible value of input parameter x (0.310 ≤ x ≤ 0.925). Furthermore, we can derive the following sum rule among the Dirac CP phase δ CP and mixing angles cos δ CP = cos 2θ 23 We note that maximal θ 23 leads to maximal Dirac CP phase δ CP . It is easy to check that the neutrino mass matrix m ν in Eq. (B.2) has the symmetry property It implies that the atmospheric angle changes from θ 23 to π/2 − θ 23 , the Dirac phase δ CP becomes π − δ CP , the Majorana CP phase β will add a negative factor and the other output parameters are kept intact under a change of the sign of the parameter η. For the fixed value of x and η, all the mixing angles, CP phases and neutrino masses are fully determined by m a and r. As an example, we shall give the predictions for x = 1 3 and η = − 4π 5 . For this case, the fixed column of PMNS matrix is 1 By comprehensively scanning over the parameter space of x, η and r, if we require the three mixing angles and mass ratio m 2 2 /m 2 3 in their 3σ regions [1], we find that all the three input parameters are restricted in narrow intervals of 0.311 ≤ x ≤ 0.381, 0.730π ≤ |η| ≤ π and 1.270 ≤ r ≤ 1.487. Furthermore, θ 12 is found to lie in a narrow interval around its 3σ upper bound 0.334 ≤ sin 2 θ 12 ≤ 0.350, and θ 23 can only take the value in the range [0.425, 0.575]. The predictions for the two CP phases δ CP and β are −0.611π ≤ δ CP ≤ −0.389π and −0.281π ≤ β ≤ 0.281π, respectively. 2 It is easy to check that the column vector (0, x, x−2) T is an eigenvector of the above neutrino mass matrix and the corresponding eigenvalue is 0. Before diagonalizing the neutrino mass matrix m ν , it is useful to perform a unitary transformation U ν1 , and this unitary transformation will make m ν to be a block diagonal matrix. Here the unitary matrix U ν1 takes the following form Then the nonzero parameters y, z and w of m ν are given by Then we need to put m ν into diagonal form with real non-negative masses, which can be done exactly by using the standard procedure shown in section 2, i.e. U T ν2 m ν U ν2 = diag(0, m 2 , m 3 ). Hence the lepton mixing matrix is determined to be 14) The lepton mixing angles and CP invariants can be read out as As a consequence, sum rules among the Dirac CP phase δ CP and the mixing angles can be found as follows , cos δ CP = cot 2θ 23 3x(x − 2) − 5x 2 − 10x + 8 cos 2θ 13 φ sol /v φs x η m a (meV) r χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 δ CP /π β/π m 2 (meV) m 3 (meV) m ee (meV) For a given value of x, the possible ranges of sin 2 θ 12 can be obtained from the above correlations by varying θ 13 over its 3σ ranges and we also can obtain the prediction for cos δ CP from the 3σ ranges of mixing angles θ 13 and θ 23 . For fixed x and η, all mixing parameters and neutrino masses depend on two input parameters m a and r. The results of the χ 2 analysis for some benchmark values of x and η are reported in table 13. From table 13 we find that the results for η = 0 are viable for both x = −8 and 10. Furthermore, maximal atmospheric mixing angle, maximal Dirac CP phase and trivial Majorana CP phase are obtained for η = 0.
The admissible range of x, r and η can be obtained from the requirement that the three mixing angles and mass ratio m 2 2 /m 2 3 are in the experimentally preferred 3σ ranges [1]. When all three mixing angles and mass ratio m 1 Subsequently a unitary transformation U ν1 is performed on the light neutrino fields, then m ν = U T ν1 m ν U ν1 becomes a block diagonal matrix with a block of 2 × 2 matrix. The unitary transformation matrix U ν1 can take the following form The parameters y, z and w in neutrino mass matrix m ν are y = − x 2 + 1 m a + 9m s e iη x 2 + 3 , z = 2(x 2 + 1) −m a + 3x 2 m s e iη x 2 + 3 , w = − 2 m a + x 4 m s e iη x 2 + 3 .

(B.19)
Then the block diagonal neutrino mass matrix m ν can be diagonalized by a unitary rotation matrix U ν2 for NO case given in Eq. (2.16). From the expressions of the matrices U l in Eq. (3.3), U ν1 and U ν2 , we find the lepton mixing matrix is The expressions for the lepton mixing angles are as follows, .
These give a sum rule between mixing angles θ 12 and θ 13 with .
On the one hand, for the fixed value of x, the possible values of θ 12 will always be limited to a narrow range by varying the mixing angle θ 13 over its 3σ range. On the other hand, from the 3σ ranges of mixing angles θ 13 and θ 12 , we find that x ≤ −4.997 or x > 5.392 should be satisfied. From the PMNS matrix, we find that the two CP rephasing invariants J CP and I 1 are predicted to be We can derive the following sum rule among the Dirac CP phase δ CP and mixing angles cos δ CP = 2 cos 2 θ 13 2 √ 3x − 3 cos 2θ 23 + x 2 (3 − 5 cos 2θ 13 ) cos 2θ 23 4 sin θ 13 sin 2θ 23 x 2 (3 − x 2 + 3 (1 + x 2 ) cos 2θ 13 ) .

(B.24)
For a given value of x, the possible ranges of cos δ CP can be obtained from the above sum rule by varying θ 13 and θ 23 over their 3σ regions. Detailed numerical analyses show that all the three mixing angles and mass ratio m 2 2 /m 2 3 can simultaneously lie in their respective 3σ ranges for input parameters |x|, η and r lying in the ranges [7.347, 9.104], [−π, π] and [0.0324, 0.0549], respectively. Then θ 12 is found to lie in a narrow interval 0.326 ≤ sin 2 θ 12 ≤ 0.330, the atmospheric mixing angle is constrained to lie in the interval 0.485 ≤ sin 2 θ 23 ≤ 0.515 and |δ CP | is predicted to be in the range of [0, 0.0190π] ∪ [0.981π, π]. Any value between −π and π is permitted for Majorana CP phase β. Here we find the Dirac CP phase is approximate trivial. Hence this breaking pattern would be ruled out if the signal of maximal δ CP is confirmed by future neutrino facilities . In table 14 we present the predictions for mixing angles, CP violating phases, light neutrino masses and the effective mass in neutrinoless double beta decay for some benchmark values of the parameters x and η. We find that the results of η = 0 are viable. It is useful in model building. However, η = 0 leads to trivial Dirac CP phase and Majorana CP phase. The reason is that parameters y, z and w in Eq. (B.19) are all real. From the expressions of parameters ψ, ρ and σ given in Eq. (2.19), we find that the three parameters can only take 0 or π. Then up to the diagonal phase matrix P ν with entries ±1 or ±i, it is easy to check that PMNS matrix in Eq. (B.20) is a real matrix. This mixing matrix gives trivial Dirac CP phase and Majorana CP phase. φ sol /v φs x η m a (meV) r χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 δ CP /π β/π m 2 (meV) m 3 (meV) m ee (meV) It is easy to check that the above neutrino mass matrix has an eigenvalue 0 with eigenvector (x, 1, 1) T . The neutrino mass matrix m ν can be block diagonalised by the unitary matrix U ν1 , where U ν1 is Then the three nonzero parameters y, z and w of m ν are x η m a (meV) r χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 δ CP /π β/π m 2 (meV) m 3 (meV) m ee (meV)   The neutrino mass matrix m ν can be diagonalized by unitary matrix U ν2 which show in Eq. (2.16). Then the lepton mixing matrix is determined to be One can straightforwardly extract the lepton mixing angles and the two CP rephasing invariants J CP and I 1 as follows, We see that the three mixing angles and Dirac CP phase only depend on two free parameters θ and ψ. Then mixing parameters are strongly correlated such that the following sum rules among the mixing angles and Dirac CP phase are found to be satisfied , cos δ CP = 3 + 6x − 5 + 2x + 2x 2 cos 2θ 13 cot 2θ 23 The former correlation implies that the solar mixing angle θ 12 is restricted by the observed value of θ 13 for a given x. From the sum rule among δ CP and mixing angles, we find that maximal atmospheric mixing angle θ 23 = 45 • leads to maximal Dirac CP phase, i.e. cos δ CP = 0. For fixed value of x, the possible value of δ CP is determined by the 3σ ranges of mixing angles θ 13 and θ 23 .
In order to see how well the lepton mixing angles can be described by this breaking pattern and its prediction for CP phases, we perform a χ 2 analysis defined in Eq. (3.5) for some benchmark values of x and η. The results are listed in table 15. We can see that the minimum value of χ 2 can be quite small for x = 4, 15/4 and 19/5. Furthermore, we also find that the mixing pattern with x = 4 is equivalent to the breaking pattern N 1 with X sol = {1, SU } and x = −1. In order to obtain all possible values of mixing angles and CP phases, we consider input parameters x, η and r being free parameters and require all three mixing angles and mass ratio m 2 2 /m 2 3 lying in their 3σ ranges. Then we find the allowed values of input parameters x, |η| and r are [3.472, 4.

C Other mixing patterns with IO
In this appendix, we shall list the other possible choices for the residual symmetries (G l , G atm , G sol and the resulting predictions for lepton mixing parameters and neutrino masses. Here the residual symmetries in the charged lepton sector, atmospheric neutrino sector and solar neutrino sector are the same as N 4 case which is discussed in section 3. Then the light neutrino mass matrix m ν takes the same form as it in Ref. [58]. From the discussing below Eq. (2.21), the PMNS matrix of this case can be directly obtained from the PMNS matrix in N 4 : The lepton mixing angles and CP rephasing invariants can be read off as Here only the residual CP transformation X sol = {U, T } can give phenomenological predictions. Then the VEV alignments of the flavons φ atm and φ sol are Form the general VEVs of flavon fields φ atm and φ sol , we find the general form of the neutrino mass matrix is given by It can be block diagonalised by the unitary matrix U ν1 , where unitary matrix U ν1 takes the following form Then the three nonzero elements y, z and w are determined to be Then neutrino mass matrix m ν can be diagonalized by performing the unitary transformation matrix U ν2 which is given in Eq. (2.16). As a consequence, the PMNS matrix is Its predictions for the three mixing angles and the two CP rephasing invariants are sin 2 θ 13 = x 2 2 + x 2 , sin 2 θ 12 = cos 2 θ , sin 2 θ 23 = 1 2 , It predicts a maximal atmospheric mixing angle θ 23 . The viable range of x can be obtained by varying θ 13 over its 3σ range, i.e. |x| ∈ [0.206, 0.225]. In order to see how well the lepton mixing angles can be described by this breaking pattern and its predictions for CP phases, we shall perform a numerical analysis. When the experimentally allowed regions at 3σ confidence level of mixing parameters and mass ratio m In this combination of residual flavor symmetries, the residual CP transformation X sol is viable only when it takes the transformations {U, ST S}. In this breaking pattern, the vacuum alignments of flavons φ atm and φ sol can be read out from table 2: The general form of the light neutrino mass matrix is determined to be (C.11) Before diagonalizing the neutrino mass matrix, we first perform a unitary U ν1 to it, where the unitary transformation matrix U ν1 takes the following form The expressions of the parameters y, z and w are The unitary transformation U ν2 diagonalizing the neutrino mass matrix m ν is of the form given in Eq. (2.16) for IO case. Then the lepton mixing matrix is determined to be of the form The mixing parameters extracted from above PMNS matrix are: Inputting the 3σ ranges of the third column of the PMNS matrix, we find the parameter (C. 16) In the case that all the three input parameters x, η and r being free parameters, we find that the three mixing angels and mass ratio m 2 1 /m 2 2 can lie in their 3σ ranges at the same time only when x, |η| and r are restricted in the range of [−10.660 In this combination of residual flavor symmetries, only the residual CP symmetry X sol = {U, ST S} is viable. From table 2, we can know the VEVs of flavons φ atm and φ sol . Then the light neutrino mass matrix takes the following form (C.17) This neutrino mass matrix can become a block diagonal matrix when we perform a unitary trans- The parameters y, z and w are take the following form The neutrino mass matrix m ν can be diagonalized by the unitary matrix U ν2 . Then the PMNS matrix is The three lepton mixing angles are predicted to be sin 2 θ 13 = 8x 2 9 + 6 √ 3x + 27x 2 , sin 2 θ 12 = cos 2 θ , sin 2 θ 23 = 1 2 , The atmospheric mixing angle is maximal and the reactor mixing angle only depends on input parameter x. Inputting the 3σ ranges of the third column of the PMNS matrix, we find the parameter This neutrino mass matrix can be diagonalized into a block diagonal form by performing a unitary transformation U ν1 to m ν , where unitary matrix U ν1 takes the following form Then the neutrino mass matrix m ν is a block diagonal matrix with elements The diagonalization matrix of m ν can be written as the form U ν2 in Eq. (2.16) for IO case. Then the PMNS matrix is determined to be Its predictions for the mixing angles and CP invariants are sin 2 θ 13 = (1 − x) 2 2 + 2x + 5x 2 , sin 2 θ 12 = cos 2 θ , sin 2 θ 23 = 1 2 , J CP = (x − 1)(1 + 2x) 2 sin 2θ cos ψ 4 (2 + 2x + 5x 2 ) 3/2 , I 1 = − (1 + 2x) 4 sin 2 2θ sin(ρ − σ) The atmospheric mixing angle θ 23 is predicted to be maximal. We find that the correct value of θ 13 can be obtained for For illustration, we shall give the χ 2 results for the typical value x = 2/3. The vacuum alignment of flavon φ sol is proportional to the column vector 1, 2 3 ω, 2 3 ω 2 T and the third column of the PMNS matrix is We see that θ 13 is rather close to its 3σ lower limit 0.2068 [1]. Hence this example should be considered as a good leading order approximation. The reason is that if subleading contributions are taken into account, accordance with experimental data are easily achieved.
Then the most general form of the light neutrino mass matrix can be easily obtained: After performing a U ν1 transformation, m ν can be a block diagonal matrix, where the unitary matrix U ν1 takes the form The nonzero elements y, z and w of the block diagonal m ν are Then the neutrino mass matrix m ν can be diagonalized by the unitary matrix U ν2 which is given in Eq. (2.16). As a consequence, the PMNS matrix can take the following form The lepton mixing parameters are predicted to be sin 2 θ 13 = x 2 1 + x 2 , sin 2 θ 12 = cos 2 θ , sin 2 θ 23 = 1 2 , J CP = x sin 2θ sin ψ 4 (1 + x 2 ) 3/2 , I 1 = − sin 2 2θ sin(ρ − σ) This mixing pattern gives a maximal θ 23 . The reactor mixing angle θ 13 only depend on parameter x which comes from the vacuum alignment of flavon φ sol . In order to obtain the value of θ 13 allowed by experiment data, the input parameter |x| must be restricted in [0.145, 0.159]. In order to accommodate the experimentally favored 3σ ranges [1] of mixing angles and mass ratio m 2 1 /m 2 2 , we find the allowed region of the parameters |η| and r are [0.0241π, 0.0346π] and [0.945, 0.954], respectively. Any values of θ 13 and θ 23 in their 3σ can be obtained. The two CP phases are predicted to be δ CP ∈ [−π, −0.988π]∪[−0.0120π, 0.00241π]∪[0.997π, π] and |β| ∈ [0.0973π, 0.111π]. Detailed numerical analyses show that accordance with experimental data can be achieved for Firstly, we perform a unitary U ν1 to m ν , where the unitary matrix U ν1 takes the following form Then the neutrino mass matrix m ν is a block diagonal matrix and the parameters y, z and w which are introduced in Eq. (2.13) are The neutrino mass matrix m ν can be diagonalized by unitary matrix U ν2 which show in Eq. (2.16). Then the lepton mixing matrix takes the following form The predictions for the lepton mixing angles as well as CP invariants are sin 2 θ 13 = (1 + 2x) 2 2 + 2x + 5x 2 , sin 2 θ 12 = cos 2 θ , sin 2 θ 23 = 1 2 , The atmospheric mixing angle θ 23 is maximal and θ 13 only depends on x. Inputting the 3σ range of θ 13 , we find the parameter Furthermore, we think it is necessary to give the predictions for three mixing angles and two CP phases. We obtain the viable ranges of mixing angles and CP phases by scanning the input parameters x, r and η in their ranges. Then we find that mixing angles θ 13 For this combination of residual symmetries, the vacuum alignments of the flavon field φ sol is Then the light neutrino mass matrix is given by In order to diagonalize this mass matrix, we first perform a unitary U ν1 to it, where the unitary matrix U ν1 is Then the light neutrino mass matrix m ν is a block diagonal matrix with the nonzero parameters y, z and w: Following the procedures presented in section 1, we know that the neutrino mass matrix m ν can be diagonalized by the unitary matrix U ν2 . Then the PMNS matrix is given by One can straightforwardly extract the lepton mixing angles and CP phases as follows sin 2 θ 13 = 1 1 + x 2 , sin 2 θ 12 = cos 2 θ, sin 2 θ 23 = 1 2 , θ 23 is predicted to be maximal and θ 13 only depends on input parameter x which comes from the general VEV invariant under the action of the residual symmetry Z T 2 U 2 × H sol CP with 3 representation. When we take into account the current 3σ bounds of θ 13 , we find that parameter |x| is constrained to be in the range of [6.293, 6.882].
In order to give the predictions for mixing parameters, we could focus on the admissible values of x, r and η in their ranges given I 1 . The admissible ranges of x, r and η can be obtained from the requirement that the three mixing angles and mass ratio m 2 1 /m 2 2 in their experimentally preferred 3σ ranges, i.e. |x| ∈ [6.293, 6.882], |η| ∈ [0.0034π, 0.0049π] and r ∈ [0.0105, 0.0120]. Then the possible ranges of the absolute values of the two CP phases δ CP and β are [0.598π, 0.661π] and [0.527π, 0.574π], respectively. This mixing pattern gives not predictions for mixing angles θ 12 and θ 13 . Here we shall give an example (x = 4 √ 3) to show how well the lepton mixing angles can be described by this mixing pattern and the predictions for CP phases. In this example, the fixed column which only depends on the VEVs of φ atm and φ sol is determined to be 1 7 1, 2 Then the best fit point and the predictions for various observable quantities obtained at the best fit point are In order to diagonalise the neutrino mass m ν , we first apply the unitary transformation U ν1 to yield m ν = U T ν1 m ν U ν1 being a block diagonal matrix, where the unitary matrix U ν1 is The parameters y, z and w of m ν is given by y = (x + 2) 2 m a − 3(x − 1) 2 m s e iη 2x 2 + 2x + 5 , z = i 3(x 2 + 2) −(x + 2)m a + (x − 1)(2x + 1)m s e iη 2x 2 + 2x + 5 , w = x 2 + 2 −3m a + (2x + 1) 2 m s e iη 2x 2 + 2x + 5 . (C.52) The nonzero parameters y, z and w in m ν are The general form of the diagonalization matrix of the neutrino mass matrix m ν is the form of U ν2 shown in Eq. (2.16) for IO case. Using the diagonalization matrix of the charged lepton mass matrix in Eq. 3.3, we find that the lepton mixing matrix is fixed to be Here the expressions of θ, ψ, ρ and σ which with respect to free parameter x, η and r are the same as them in Eq. (B.14). Furthermore, the expressions of the neutrino masses m 1 and m 2 are the same as the expressions of m 2 and m 3 in the case of N 7 , respectively. For the mixing matrix in Eq. (C.63), we can obtain the following expressions for the mixing angles and the CP invariants , sin 2 θ 12 = sin 2 θ , sin 2 θ 23 = 1 2 , We find that the atmospheric mixing angle θ 23 is 45 • . The experimentally allowed region of x depends of the 3σ range of θ 13 . We find the viable range of We note that the best fit value of sin 2 θ 13 is rather close to its 3σ upper limit 0.02463. Hence we think that this breaking pattern with x = 4 5 is a good leading order approximation. Furthermore we perform a comprehensive numerical analysis. When three mixing angles and mass ratio Then the predictions for the three mixing angles and two CP invariants are , sin 2 θ 12 = sin 2 θ , 36 (x 2 + 1) 2 . (C.67) From Eq. (C.66), we find that the third column of PMNS matrix only depends on the parameter x which dictates the vacuum alignment of flavon φ sol . Since both mixing angles θ 13 and θ 23 depend on only one input parameter x. Then we can obtain the following sum rule sin 2 θ 23 = 1 2 ± tan θ 13 2 2 − tan 2 θ 13 1 2 ± √ 2 tan θ 13 2 , (C. 68) where "+" sign in ± is satisfied for x > 0 and "−" for x < 0. Given the experimental 3σ range of θ 13 , we have 0.602 ≤ sin 2 θ 23 ≤ 0.612 or 0.388 ≤ sin 2 θ 23 ≤ 0.398. The later range has been removed by the 3σ range of θ 23 . The experimental data of the third column of PMNS matrix at 3σ level can be accommodated for the parameter x with x ∈ [0.179, 0.196]. The requirement with three mixing angles and mass ratio m 2 1 /m 2 2 in their 3σ ranges require the other two input parameters |η| and r in the ranges of [0.9929π, 0.9950π] and [0.3226, 0.3248], respectively. Then the mixing angle θ 23 and two CP phases are predicted to be 0.602 ≤ sin 2 θ 23 ≤ 0.612, 0.580π ≤ |δ CP | ≤ 0.628π and 0.431π ≤ |β| ≤ 0.469π. The other two mixing angles can take any values in their 3σ ranges. Now let us give the numerical results of a relatively simple example with x = 1 3 √ 3 . In this example, the third column of the PMNS matrix is 1 √ 42 (1, 5, 4) T which is agrees with all measurements to date [1]. When we perform a χ 2 analysis, the predictions for various observable quantities are We firstly perform the following unitary transformation to light neutrino fields Then the neutrino mass matrix m ν = U T ν1 m ν U ν1 is a block diagonal matrix with nonzero elements y = 2m a − 3 2 x 2 m s e iη , z = − i 2 x 3(2 + x 2 )m s e iη , w = 1 2 2 + x 2 m s e iη . (C.72) The most general diagonalization matrix of m ν is the unitary matrix U ν2 which is given in Eq. (2.16). We can obtain the PMNS matrix taking the following form We see that the best fit value of θ 13 is a bit smaller than its 3σ lower limit 0.02068 [1]. We think it is a good leading order approximation. (1 + 2xi) 2 1 (C.76) In order to diagonalize m ν , we first perform a unitary transformation U ν1 to m ν , where the unitary matrix U ν1 is When we perform the unitary transformation U ν1 to m ν , we obtain a block diagonal m ν with parameters y, z and w being y = 3m a − 3x 2 m s e iη , z = −3ix 1 + x 2 m s e iη , w = 3 1 + x 2 m s e iη . (C.78) The diagonalization matrix of block diagonal m ν can take to be U ν2 which is given in Eq. (2.16). As a consequence, the PMNS matrix can take the following form The lepton mixing angles and CP phases are found to be of the form sin 2 θ 13 = 1 1 + x 2 , sin 2 θ 12 = cos 2 θ, sin 2 θ 23 = 1 2 , J CP = − x 2 sin 2θ sin ψ 4 (1 + x 2 ) 3/2 , I 1 = − x 4 sin 2 2θ sin(ρ − σ) 4 (1 + x 2 ) 2 .
(C. 80) We see that θ 23 is maximal. In order to obtain viable θ 13 , the absolute value of the input parameter x must lie in the range of [6.293, 6.882]. Freely varying the three mixing angles and the mass ratio m 2 1 /m 2 2 in their 3σ ranges, we find that other input parameters |η| and r are limited in the range [0.0033π, 0.0049π] and [0.0104, 0.0124], respectively. Moreover, we find the values of CP phases |δ CP | and |β| are in the intervals [0.839π, 0.905π] ∪ [0.0955π, 0.162π] and [0.526π, 0.574π], respectively. Any value in the 3σ range of θ 12 can be taken in this mixing pattern. For model building convenience, we shall give the analysis for x = −7. In this example, the third column of PMNS matrix is determined to (C.82) This neutrino mass matrix can be diagonalized to a block diagonal matrix when we perform a unitary transformation U ν1 , where the unitary matrix U ν1 is Then we can obtain the three parameters y, z and w in m ν are y = 3x 2 + 1 m a − 3m s e iη 2x 2 + 1 , z = 3x 2 + 1 (−2ix 2 + x − i)m a + 3x(4x 2 + 1)m s e iη (2x 2 + 1) (3x 2 + 1)(4x 2 + 1) , w = − (2x 2 + ix + 1) 2 m a + 3x 2 (4x 2 + 1) 2 m s e iη (2x 2 + 1)(4x 2 + 1) . (C.84) As a result, m ν can be diagonalized by the unitary matrix U ν2 which is given in Eq. (2.16). From expressions of parameters y, z and w in Eq. (C.84), the absolute values and the phases of them can be obtained. Then from Eq. (2.17), we can get the neutrino masses m 2 1 and m 2 2 . Following the procedures listed in section 2, the expressions of sin 2θ, cos 2θ, sin ψ, cos ψ, sin ρ, cos ρ, sin σ and cos σ can be obtained. The lepton mixing matrix read as It is straightforward to extract the mixing angles and the two CP rephasing invariants sin 2 θ 13 = x 2 1 + 3x 2 , sin 2 θ 12 = sin 2 θ , sin 2 θ 23 = 1 2 + 4x 2 , J CP = x √ 1 + 4x 2 sin 2θ sin ψ 4(1 + 3x 2 ) 3/2 , I 1 = − 2x 2 + 1 2 sin 2 2θ sin(ρ − σ) 4 (3x 2 + 1) 2 . (C.86) We see that both the atmospheric mixing angle and the reactor mixing angle only depend on one input parameter x which decides the vacuum alignment of flavon φ sol . Then a sum rule between mixing angle θ 13 and θ 23 is obtained This sum rule has also been obtained in Ref. [37]. It implies that the atmospheric mixing angle is in the first octant, i.e. θ 23 < 45 • . For the fitted 3σ range of θ 13 , the atmospheric mixing angles is constrained to be in the interval of 0.475 ≤ sin 2 θ 23 ≤ 0.479. This can be tested in future neutrino oscillation experiments. Inserting the 3σ ranges of θ 13 , we find the viable range of |x| is (1 + 2ix) 2 1 + ix + 2x 2 ω 1 + ix + 2x 2 ω 2 (C.89) In order to diagonalize the light neutrino mass matrix m ν , it is useful to first perform a unitary transformation U ν1 . Here the unitary matrix U ν1 takes to be Then the neutrino mass matrix m ν is a block diagonal matrix with elements y = 3x 2 + 1 m a + 3x 2 m s e iη 2x 2 + 1 , z = (2x − i) 3x 2 + 1 (1 − ix)m a + 3x ix 2 + x + i m s e iη (2x 2 + 1) (3x 2 + 1)(4x 2 + 1) , w = (i − 2x) (x + i) 2 m a + 3(x 2 − ix + 1) 2 m s e iη (2x + i) (2x 2 + 1) . (C.91) Then we can obtain the absolute values and the phases of parameters y, z and w. As a result, m ν can be diagonalized by the unitary matrix U ν2 which is given in Eq. (2.16). It is easily to get the neutrino masses m 2 1 and m 2 2 by inserting the parameters y, z and w into Eq. (2.17). Then the parameters θ, ψ, ρ and σ can be given in dependence on the absolute values and the phases of parameters y, z and w. Form the matrix form of U l in Eq. (3.3), U ν1 and U ν2 , we find that the lepton mixing matrix is given by x cos θ √ (2x 2 +1)(3x 2 +1) (C.92) Then the lepton mixing parameters are predicted to be sin 2 θ 13 = x 2 1 + 3x 2 , sin 2 θ 12 = sin 2 θ , sin 2 θ 23 = 1 − 1 2 (1 + 2x 2 ) , J CP = − x √ 1 + 4x 2 sin 2θ sin ψ 4(1 + 3x 2 ) 3/2 , I 1 = − 2x 2 + 1 2 sin 2 2θ sin(ρ − σ) 4 (3x 2 + 1) 2 . (C.93) Both the atmospheric mixing angle and the reactor mixing angle depend on only one input parameter x which comes from the vacuum alignment invariant under the action of the residual symmetry in the solar neutrino sector. As a consequence, the following sum rule between the reactor mixing angle and the atmospheric mixing angle is found to be satisfied sin 2 θ 23 = 1 2 + tan 2 θ 13 . (C.94) We note that θ 23 is constrained to lie in the second octant. Inputting the experimentally preferred 3σ range of θ 13 , the atmospheric mixing angle is predicted to be 0.521 ≤ sin 2 θ 23 ≤ 0.525. It is remarkable that a good fit to the experimental data can always be achieved for any |x|. When these two mixing angles are required in their 3σ, the input parameter x is constricted in the range of [0.148, 0.163]. Similar to the example in I 17 , we also give the example of x = The input parameters x, r and η are treated as random numbers, and mixing angles and mass ratio are required in their 3σ ranges. We find the allowed regions of the parameter |η| and r are [0.0905π, 0.108π] and [0.6328, 0.6448], respectively. The predicted range of the two CP phases are predicted to be |δ CP | ∈ [0, 0.0619π] ∪ [0.710π, 0.777π] and |β| ∈ [0.485π, 0.536π].